In this paper we shall discuss the problem how to use a priori information for constructing regularizing algorithms and error estimation while solving ill-posed problems. We shall consider the following types of a priori information: (1) a compactness of a set of solutions; (2) a sourcewise representation of a solution with a compact operator.

A new scheme of a posteriori accuracy estimation for approximate solutions to ill-posed inverse problems is presented along with an algorithm of calculating this estimation. A new notion of extra-optimal regularizing algorithmis introduced as a method for solving ill-posed inverse problems having optimal in order a posteriori accuracy estimation. Sufficient conditions of extra-optimality are formulated and an example of extra-optimal regularizing algorithm is given. The developed theory is illustrated by numerical experiments.

: The problem of a posteriori accuracy estimation for approximate solutions of ill-posed extremal problem is under investigation. We present a new approach to a posteriori estimation along with the study of principal properties of the used estimator. The numerical algorithm for the solution of central extremal problem of a posteriori accuracy estimation is formulated as well. We analyze the optimality in order of obtained a posteriori accuracy estimates and introduce a new important notion of extra-optimal regularizing algorithm as a method for the solution of ill-posed problems having optimal in order a posteriori accuracy estimate. Finally, we demonstrate the results of numerical experiments on applications of extra-optimal regularizing methods, constructed on the base of Tikhonov and Lavrent'ev regularization, for calculating a posteriori accuracy estimates.

Two modifications with variable coefficients of the well-known SEIR model for epidemic development in the application to the modeling of the infection curves of COVID-19 are considered. The data for these models are information on the number of infections each day obtained from the Johns Hopkins Coronavirus Resource Center database. In our paper, we propose special methods based on Tikhonov regularization for models’ identification on the class of piecewise constant coefficients. In contrast to the model with constant coefficients, which cannot always accurately describe some of infection curves, the first model is able to approximate them for different countries with an accuracy of 2–8%. The second model considered in the article takes into account external sources of infection in the form of an inhomogeneous term in one of the model equations and is able to approximate the data with a slightly better accuracy of 2–4%. For the second model, we also consider the possibility of using other input data, namely the number of infected people per day. Such data are used to model infection curves for several waves of the COVID-19 epidemic, including part of the Omicron wave. Numerical experiments carried out for a number of countries show that the waves of external sources of infection found are ahead of the wave of infection by 10 or more days. At the same time, other piecewise constant coefficients of the model change relatively slowly. These models can be applied fairly reliably to approximate many waves of infection curves with high precision and can be used to identify external and hidden sources of infection. This is the advantage of our models.